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Properties

Label 430.2.l.a
Level $430$
Weight $2$
Character orbit 430.l
Analytic conductor $3.434$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [430,2,Mod(7,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.l (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{6} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{6} + 12 q^{7} + 16 q^{13} - 88 q^{16} + 4 q^{17} - 16 q^{21} + 12 q^{23} - 8 q^{25} + 12 q^{28} - 36 q^{30} - 40 q^{31} + 12 q^{33} - 40 q^{35} - 40 q^{36} + 16 q^{38} - 56 q^{41} - 56 q^{43} + 24 q^{46} + 72 q^{47} + 24 q^{50} - 16 q^{52} + 20 q^{53} + 24 q^{55} - 8 q^{56} + 20 q^{57} + 8 q^{60} + 72 q^{61} + 36 q^{62} - 40 q^{66} - 16 q^{67} + 4 q^{68} + 24 q^{71} + 60 q^{73} - 48 q^{76} - 48 q^{77} + 40 q^{78} + 28 q^{81} + 20 q^{86} - 64 q^{87} - 8 q^{90} + 48 q^{91} - 12 q^{92} - 108 q^{93} + 4 q^{95} - 4 q^{96} - 48 q^{97} - 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.707107 + 0.707107i −0.695712 + 2.59643i − 1.00000i −0.0995241 − 2.23385i −1.34401 − 2.32790i 0.420775 + 1.57036i 0.707107 + 0.707107i −3.65937 − 2.11274i 1.64995 + 1.50920i
7.2 −0.707107 + 0.707107i −0.547703 + 2.04406i − 1.00000i 2.11862 − 0.715150i −1.05808 − 1.83265i −0.838501 − 3.12933i 0.707107 + 0.707107i −1.28011 − 0.739071i −0.992405 + 2.00378i
7.3 −0.707107 + 0.707107i −0.424244 + 1.58330i − 1.00000i −1.40555 − 1.73909i −0.819577 − 1.41955i −0.632122 − 2.35911i 0.707107 + 0.707107i 0.271217 + 0.156587i 2.22359 + 0.235851i
7.4 −0.707107 + 0.707107i −0.388695 + 1.45063i − 1.00000i −0.951195 + 2.02367i −0.750902 − 1.30060i 0.957274 + 3.57260i 0.707107 + 0.707107i 0.644831 + 0.372294i −0.758352 − 2.10355i
7.5 −0.707107 + 0.707107i −0.348214 + 1.29955i − 1.00000i 1.37552 + 1.76294i −0.672697 − 1.16515i 0.277286 + 1.03484i 0.707107 + 0.707107i 1.03050 + 0.594958i −2.21922 − 0.273949i
7.6 −0.707107 + 0.707107i 0.0121292 − 0.0452669i − 1.00000i −2.07202 + 0.840686i 0.0234319 + 0.0405852i −0.477251 − 1.78112i 0.707107 + 0.707107i 2.59617 + 1.49890i 0.870681 − 2.05959i
7.7 −0.707107 + 0.707107i 0.231077 − 0.862392i − 1.00000i 2.23333 − 0.110550i 0.446407 + 0.773199i 0.738092 + 2.75460i 0.707107 + 0.707107i 1.90775 + 1.10144i −1.50103 + 1.65738i
7.8 −0.707107 + 0.707107i 0.392168 − 1.46359i − 1.00000i −2.07474 − 0.833933i 0.757610 + 1.31222i 1.23476 + 4.60819i 0.707107 + 0.707107i 0.609775 + 0.352054i 2.05674 − 0.877385i
7.9 −0.707107 + 0.707107i 0.506805 − 1.89142i − 1.00000i 0.195585 − 2.22750i 0.979072 + 1.69580i −0.522322 − 1.94933i 0.707107 + 0.707107i −0.722552 − 0.417165i 1.43678 + 1.71338i
7.10 −0.707107 + 0.707107i 0.653568 − 2.43915i − 1.00000i 1.81492 + 1.30616i 1.26260 + 2.18688i −0.0630011 − 0.235123i 0.707107 + 0.707107i −2.92423 − 1.68830i −2.20694 + 0.359750i
7.11 −0.707107 + 0.707107i 0.867639 − 3.23807i − 1.00000i −1.13496 + 1.92662i 1.67615 + 2.90318i 0.0566213 + 0.211314i 0.707107 + 0.707107i −7.13424 − 4.11896i −0.559792 − 2.16486i
7.12 0.707107 − 0.707107i −0.833140 + 3.10932i − 1.00000i 1.35223 + 1.78086i 1.60950 + 2.78774i 1.22409 + 4.56838i −0.707107 − 0.707107i −6.37568 − 3.68100i 2.21543 + 0.303091i
7.13 0.707107 − 0.707107i −0.727882 + 2.71649i − 1.00000i −2.12803 − 0.686647i 1.40616 + 2.43554i −0.979873 − 3.65693i −0.707107 − 0.707107i −4.25143 − 2.45457i −1.99028 + 1.01921i
7.14 0.707107 − 0.707107i −0.426111 + 1.59027i − 1.00000i 1.81858 + 1.30107i 0.823184 + 1.42580i −0.688340 − 2.56892i −0.707107 − 0.707107i 0.250692 + 0.144737i 2.20592 − 0.365937i
7.15 0.707107 − 0.707107i −0.356683 + 1.33116i − 1.00000i −2.17879 − 0.502846i 0.689058 + 1.19348i 0.982140 + 3.66540i −0.707107 − 0.707107i 0.953316 + 0.550397i −1.89621 + 1.18507i
7.16 0.707107 − 0.707107i −0.103919 + 0.387831i − 1.00000i 1.79733 − 1.33027i 0.200756 + 0.347719i −0.783039 − 2.92234i −0.707107 − 0.707107i 2.45846 + 1.41939i 0.330259 − 2.21154i
7.17 0.707107 − 0.707107i −0.00223075 + 0.00832526i − 1.00000i −1.58545 + 1.57682i 0.00430947 + 0.00746423i 0.306192 + 1.14273i −0.707107 − 0.707107i 2.59801 + 1.49996i −0.00610084 + 2.23606i
7.18 0.707107 − 0.707107i 0.0996077 − 0.371741i − 1.00000i 2.14837 − 0.620072i −0.192427 − 0.333294i 1.10238 + 4.11412i −0.707107 − 0.707107i 2.46981 + 1.42594i 1.08067 − 1.95759i
7.19 0.707107 − 0.707107i 0.118826 − 0.443464i − 1.00000i −0.834055 − 2.07469i −0.229554 − 0.397599i −0.174257 − 0.650336i −0.707107 − 0.707107i 2.41554 + 1.39461i −2.05680 − 0.877264i
7.20 0.707107 − 0.707107i 0.555341 − 2.07256i − 1.00000i −1.79466 + 1.33387i −1.07284 − 1.85821i −1.21892 − 4.54906i −0.707107 − 0.707107i −1.38903 − 0.801955i −0.325825 + 2.21220i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.d odd 6 1 inner
215.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.l.a ✓ 88
5.c odd 4 1 inner 430.2.l.a ✓ 88
43.d odd 6 1 inner 430.2.l.a ✓ 88
215.l even 12 1 inner 430.2.l.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.l.a ✓ 88 1.a even 1 1 trivial
430.2.l.a ✓ 88 5.c odd 4 1 inner
430.2.l.a ✓ 88 43.d odd 6 1 inner
430.2.l.a ✓ 88 215.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).